Geometric, Electronic, and Optoelectronic Properties of Carbon-Based Polynuclear C3O[C(CN)2]2M3 (where M = Li, Na, and K) Clusters: A DFT Study

Carbon-based polynuclear clusters are designed and investigated for geometric, electronic, and nonlinear optical (NLO) properties at the CAM-B3LYP/6-311++G(d,p) level of theory. Significant binding energies per atom (ranging from −162.4 to −160.0 kcal mol−1) indicate excellent thermodynamic stabilities of these polynuclear clusters. The frontier molecular orbital (FMOs) analysis indicates excess electron nature of the clusters with low ionization potential, suggesting that they are alkali-like. The decreased energy gaps (EH-L) with increased alkali metals size revael the improved electrical conductivity (σ). The total density of state (TDOS) study reveals the alkali metals’ size-dependent electronic and conductive properties. The significant first and second hyperpolarizabilities are observed up to 5.78 × 103 and 5.55 × 106 au, respectively. The βo response shows dependence on the size of alkali metals. Furthermore, the absorption study shows transparency of these clusters in the deep-UV, and absorptions are observed at longer wavelengths (redshifted). The optical gaps from TD-DFT are considerably smaller than those of HOMO-LUMO gaps. The significant scattering hyperpolarizability (βHRS) value (1.62 × 104) is calculated for the C3 cluster, where octupolar contribution to βHRS is 92%. The dynamic first hyperpolarizability β(ω) is more pronounced for the EOPE effect at 532 nm, whereas SHG has notable values for second hyperpolarizability γ(ω).


Introduction
Nonlinear optical (NLO) materials are at the front line of research in interdisciplinary science and laser-based technology due to their fundamental applications in the field of optoelectronics [1][2][3][4]. Photonic devices, laser-based technology, endoscope, and sensors are examples of well known technologies where NLO materials have possible applications [5][6][7][8][9][10][11][12]. To design and synthesize the NLO materials, much efforts are exerted to understand the origin of nonlinearity in molecules and clusters in order to correlate NLO responses to electronic structure and molecular geometry. Polarization, asymmetric charge distribution, asymmetric crystal packing, and π-conjugated electron transport routes are all required for NLO materials. Because of their high thermal stability and transparency, inorganic nonlinear optical materials have been preferred over organic ones [13]. Some inorganic borates crystals, such as KB 5 (KB 5 O 8 H 2 O), BBO (-BaB 2 O 4 ), and LiB 3 O 5 (LBO), have been investigated as good NLO materials, particularly in the ultraviolet range [13].
For obtaining high-performance NLO materials, several strategies were proposed, which include bond length alternation (BLA) [14], doping metal atoms [15], push-pull  The thermodynamic stability of the studied polynuclear clusters is evaluated through calculated binding energy per atom (E b ). Overall, the binding energies range from −160.1 to −162.1 kcal mol −1 (Table 1), where the highest energy is found for C1, while the lowest is observed for the C2 cluster. The obtained significant binding energies per atom suggests their thermodynamic stabilities. The calculated binding energies are higher in comparison to previously reported superalkali clusters NM'M (where M = Li, Na and K), C 3 X 3 Y 3 (X = O, S, and Y = Li, Na and K) and bimetallic superalkali clusters [47,48]. The trend of binding energies per atom for studied clusters is also shown in Figure 2. Compared to clusters C2 and C3, cluster C1 has a greater binding energy value. The computed binding energies show high thermal stability of these clusters, which demonstrate that they can be synthesized experimentally.

Electronic Properties and Stability
The electronic stability and superalkali nature of these clusters can be observed from calculated ionization potential and electron affinity. The obtained vertical ionization potential values are smaller than Cs-atom (3.89 eV), which shows the superalkali characteristics of these clusters. These values are also significant and account for the electronic stabilities of these clusters. The highest VIE value of 3.65 eV is found for C1, while the lowest value (3.0 eV) is indicated for the C3 cluster (Table 1). A gradual decrease in VIE values with the increased size of alkali metals can be seen in these clusters. On the other hand, the vertical electron affinity (VEA) values range from 0.27 to 0.89 eV, where C3 shows the lowest value. The reduced values of EA indicate the electropositive nature of these clusters.

Electronic Properties and Stability
The electronic stability and superalkali nature of these clusters can be observed from calculated ionization potential and electron affinity. The obtained vertical ionization potential values are smaller than Cs-atom (3.89 eV), which shows the superalkali characteristics of these clusters. These values are also significant and account for the electronic stabilities of these clusters. The highest VIE value of 3.65 eV is found for C1, while the lowest value (3.0 eV) is indicated for the C3 cluster (Table 1). A gradual decrease in VIE values with the increased size of alkali metals can be seen in these clusters. On the other hand, the vertical electron affinity (VEA) values range from 0.27 to 0.89 eV, where C3 shows the lowest value.
The reduced values of EA indicate the electropositive nature of these clusters.
To obtain reactivity and charge distribution, the computed NBO charges are given in Table 1. The NBO charges (positive) on alkali metals slightly increase from Li to K metals. The charge is transferred from alkali metals to electronegative atoms (oxygen and nitrogen) within clusters. The NBO charges on alkali metals (QM) lie in the range of 0.58 to 0.62 e, where C1 shows higher charge (positive magnitude) on metals. The charge transferred from alkali to O-atom is more pronounced as compared to the alkali to N-atom transition, which may be attributed to the higher electronegativity of the oxygen atom. The calculated NBO charges on O-atom (QO) lie in the range of −0.83 to −0.96 e and are higher for small-sized metals.

Global Reactivity Descriptor
To characterize the reactivity of these clusters, we calculated global reactivity descriptor, chemical hardness, softness, and chemical potential ( Table 2). The chemical hardness is measured as resistance to change in electronic distribution within clusters. The results obtained show that the C3 cluster has the highest value (1.839 eV) of hardness, whereas the C1 has the lowest value. The size of alkali metals is an obvious factor in controlling the hardness of clusters. The decreased values show a correlation with increased atomic size (Li to K), which guarantees soft nature and reactivity (Table 2). Similarly, the values of chemical softness (S) increase from C1 to C3 and reach the maximum of 0.33 eV. The chemical potential values are also calculated and given in Table 2. The higher chemical potential (χ) values show the escaping tendency of the electrons in clusters and molecules. Obtained significant values (negative) indicate the stability of these polynuclear clusters. These values also suggest that the clusters do not decompose spontaneously into atoms and possess reasonable electronic stability.

FMO Analysis and Excess Electron Nature of Clusters
To provide deep insight into the electronic structures of the studied clusters, the densities of the highest occupied molecular orbitals (HOMO) and virtual orbitals are plotted, and their energy values are given in Table 2. The HOMO and LUMO are quite important in quantum chemistry, as they allow the prediction of chemical stability and reactivity of molecules. Imperatively, the small difference between HOMO-LUMO (E H-L ) is crucial for the description of reactivity of molecules. The smaller E H-L gaps depict greater chemical reactivity with a high tendency to be polarized, as well as low kinetic stability. The HOMO-LUMO gap values lie in the range of 4.08 to 1.96 eV, where the highest value corresponds to C1 clusters, while the lowest values correspond for C3. One can note that E H-L decreases with increased metals size (Li to K) within clusters. Furthermore, decreased E H-L gaps for the studied clusters can be attributed to increased energies of occupied orbitals where the energy of virtual orbitals goes on decreasing.
The reactivity and conducting qualities of these clusters are revealed by a significant reduction in HOMO-LUMO gaps; there are excitable valence electrons (excess electrons) with transition HOMO → LUMO. The excess electron nature is further justified by the distribution of HOMO densities, and the electronic density cloud is mainly spread over alkali metals, which indicates the excess electron character of these superalkali clusters. The three-dimensional HOMO density of C1 is shaped as a s-orbital, while for C2 and C3, its look like a diffuse p-orbital ( Figure 3). The LUMO densities that are generated are spherical and resemble s-orbitals.
its look like a diffuse p-orbital ( Figure 3). The LUMO densities that are generated ar spherical and resemble s-orbitals.

Electrical Conductivity (σ)
The electrical conductivity is also a crucial aspect to demonstrate the NLO propertie of molecules. The electrical conductivity (σ) is the function of energy gaps (EH-L); thus narrowing HOMO-LUMO gaps more will lead to higher electrical conductivity of mate rials. In our designed clusters, the HOMO-LUMO gaps are significantly reduced from 4.08 to 1.96 eV. The electrical conductivity increases with increased size of alkali metals which might be attributed to ease in excitation of electrons (HOMO to LUMO).

TD-DFT Analysis
In the transparent region, the applications of nonlinear optical materials can be bette understood. The obtained TD-DFT parameters of crucial transitions and first allowed transitions are given in Table 3. The percentage contribution of particular orbitals of thes clusters for both transitions are also given in Table 3, whereas spectra are shown in Figur 4. The higher value of ϵ shows strong absorption at particular wavelength. Additionally a higher value of fo reveals the strong transition probability. The studied cluster C3 ha significant value of ϵ and oscillator strength at higher wavelength. The absorption maxim (λmax) during crucial transition for C1, C2, and C3 are 758, 688, and 995 nm, respectively where the redshifted (i.e., bathochromic sift) in λmax is observed for C3 (Table 3). The ob tained excitation energies of crucial transition are 1.63, 0.92, and 1.24 eV for C1, C2, and C3 clusters. On the other hand, the obtained optical gaps during allowed transitions ar 1.63, 0.92, and 0.86 eV. The C1 cluster has same value for crucial excitation and optica

Electrical Conductivity (σ)
The electrical conductivity is also a crucial aspect to demonstrate the NLO properties of molecules. The electrical conductivity (σ) is the function of energy gaps (E H-L ); thus, narrowing HOMO-LUMO gaps more will lead to higher electrical conductivity of materials. In our designed clusters, the HOMO-LUMO gaps are significantly reduced from 4.08 to 1.96 eV. The electrical conductivity increases with increased size of alkali metals, which might be attributed to ease in excitation of electrons (HOMO to LUMO).

TD-DFT Analysis
In the transparent region, the applications of nonlinear optical materials can be better understood. The obtained TD-DFT parameters of crucial transitions and first allowed transitions are given in Table 3. The percentage contribution of particular orbitals of these clusters for both transitions are also given in Table 3, whereas spectra are shown in Figure 4. The higher value of shows strong absorption at particular wavelength. Additionally, a higher value of f o reveals the strong transition probability. The studied cluster C3 has significant value of and oscillator strength at higher wavelength. The absorption maxima (λ max ) during crucial transition for C1, C2, and C3 are 758, 688, and 995 nm, respectively, where the redshifted (i.e., bathochromic sift) in λ max is observed for C3 (Table 3). The obtained excitation energies of crucial transition are 1.63, 0.92, and 1.24 eV for C1, C2, and C3 clusters. On the other hand, the obtained optical gaps during allowed transitions are 1.63, 0.92, and 0.86 eV. The C1 cluster has same value for crucial excitation and optical gap, while for C2 and C3, optical gaps (allowed transition) values are significantly reduced. The excitation energies of allowed transition are decreasing monotonically from C1 to C3 with increased metal size (Li to K). The absorption maxima (λ max ) of allowed transition are observed at longer wavelength as compared to absorption during crucial transition. As a result, bigger alkali metals have a stronger influence on absorptions shift to higher wavelengths. Furthermore, these clusters are completely transparent under the deep-UV region and have broadband absorption in the near-Visible region (Figure 4). The highest energy state TD-DFT parameters also reveal transparency in the deep-UV region, while absorption is mostly in the UV-visible region (Table 3). Likewise, the gradual increase in oscillator strength (f o ) can also be seen for C1 to C3 clusters for crucial transition and allowed transition, which suggest increased quantum chemical excitation probabilities in higher-sized clusters. gap, while for C2 and C3, optical gaps (allowed transition) values are significantly reduced. The excitation energies of allowed transition are decreasing monotonically from C1 to C3 with increased metal size (Li to K). The absorption maxima (λmax) of allowed transition are observed at longer wavelength as compared to absorption during crucial transition. As a result, bigger alkali metals have a stronger influence on absorptions shift to higher wavelengths. Furthermore, these clusters are completely transparent under the deep-UV region and have broadband absorption in the near-Visible region (Figure 4). The highest energy state TD-DFT parameters also reveal transparency in the deep-UV region, while absorption is mostly in the UV-visible region (Table 3). Likewise, the gradual increase in oscillator strength (fo) can also be seen for C1 to C3 clusters for crucial transition and allowed transition, which suggest increased quantum chemical excitation probabilities in higher-sized clusters.  Figure 4. Absorption spectra of C1 to C3 clusters.

Dipole Moment (µ o ) and Change in Dipole Moment (∆µ)
For better comprehension of the electronic properties in these clusters, the dipole moment (µ o ) and change in dipole moments (∆µ) values are also calculated. Overall, the dipole moment and change in dipole moment (∆µ) values are quite significant, which reveal asymmetric electronic distribution in these clusters ( Table 4). The measured total dipole moment indicate polarity in clusters and the values of µ o are significant and range from 1.49 to 4.11 au, where the highest value is observed for the C3 cluster. On the other hand, the total change in dipole moment (∆µ) values are slightly smaller than those of dipole moment, but C2 shows a significant value of 4.69 au (Table 4).

Linear and Nonlinear Optical (NLO) Properties
To investigate the influence of excess electrons on triggering the NLO properties of studied polynuclear clusters, hyperpolarizability (β o ) and second hyperpolarizability (γ o ) are two crucial evaluation indices. The presence of excess electrons greatly increases the hyperpolarizability and second hyperpolarizability values, as shown in a number of studies [43,[47][48][49][50][51][52][53][54]. We are interested in exploring the role of excess electrons in decreasing excitation energies, which ultimately escalates hyperpolarizabilities. The calculated linear and NLO parameters for the C 3 O[C(CN) 2 ] 2 M 3 (where M = Li, Na and K) at CAM-B3LYP/6-311++g(d,p) clusters are given in Table 4. The α o values lie in the range of 2.5 × 10 2 to 6.62 × 10 2 , and there is a slight increase with the increased size of alkali metals. These values show liner optical properties of polynuclear clusters, and the presence of polarizabilities is due to asymmetric electronic density distribution in these clusters.
The hyperpolarizability values of studied clusters range from 2.37 × 10 3 to 5.78 × 10 3 au, where the highest value is obtained for C3, while the lowest value is for the C1 cluster. β o values are increasing from Li to K metals within these clusters, which shows size dependence. It can be seen that electronic properties significantly contribute to hyperpolarizability response, and the larger the change in dipole moment, the higher the hyperpolarizabilities are. Thus, β o values follow the increasing trend in these clusters, C1 < C2 < C3. Furthermore, the increased β o values have a good match with reduced ionization potential and HOMO-LUMO gaps. The trend of size-dependent β o is also shown in Figure 5.
In addition, the static second hyperpolarizability (γ o ) values are also calculated and lie in the range of 2.9 × 10 5 to 5.5 × 10 6 au (Table 4). Overall, γ o values are significant where the highest value (5.5 × 10 6 au) is obtained for the C1 cluster, while the lowest is for C3. It is found that, with the increased size of alkali metals, the γ o values decrease slightly from Furthermore, the βvec values are strongly correlated with total hyperpolarizability (βo). The calculated βvec values are given in Table 4. These values range from 2.37 × 10 3 to 5.78 × 10 3 au. The βvec is the projection of hyperpolarizability on dipole moment vector and shows close resemblance βo. However, good agreement between βo and βvec shows that the direction of the dipole moment vector and the projection of hyperpolarizability are in the same direction. The factor affecting βvec values might be the same for βo, where the highest βvec values are obtained for higher-sized alkali metals (Table 4).

Scattering Hyperpolarizability (βHRS)
Density functional theory calculations have been carried out to find scattering hyperpolarizability (βHRS), and values range from 1.34 × 10 3 to 1.62 × 10 4 au, where values are increasing steadily from the C1 to C3 cluster. The computed highest value is (1.62 × 10 4 au), found for C1 cluster, whereas the lowest value of 1.34 × 10 3 au is for the C1 cluster ( Table 4). The βHRS is the most viable parameter to calculate the hyperpolarizability of centrosymmetric molecules and clusters, even with zero change in dipole moment. There is an excellent agreement of βHRS with βo where the βHRS show dependence on the size of alkali metals (M). The increased size of alkali metals (Li to K) favors the excellent electronic properties. Therefore, it also causes significantly enhanced βHRS values. Additionally, average dipolar and octupolar hyperpolarizability, which are more prominent in C2 and C3 clusters, provide a notable contribution to βHRS. Moreover, these clusters are of octupolar molecules, which can be seen by their highest octupolar contribution Φβ(j = 3) of 92 % for C3 (Table 4).

Frequency Dependent NLO Properties
We theoretically examined the incident-frequency (ω) effect on the first and second hyperpolarizability at applied frequencies of 532 and 1064 nm. The frequency-dependent first hyperpolarizability β(ω) is calculated with the electro-optical Pockel's effect (EOPE) and second harmonic generation (SHG), whereas the γ(ω) is expressed in terms of dc-Kerr effect and second harmonic generation (SHG). Overall, the dynamic hyperpolarizabilities values are higher than those of static hyperpolarizabilities. The significant EOPE effect β(−ω; ω,0) was observed for the C3 cluster at 532 nm, while its SHG value increased up to 1.7 × 10 6 au (Table 5). It can be demonstrated that the dynamic hyperpolarizabilities are higher at the smaller incident frequency (ω = 532 nm) and slightly decreased at the higher Furthermore, the β vec values are strongly correlated with total hyperpolarizability (β o ). The calculated β vec values are given in Table 4. These values range from 2.37 × 10 3 to 5.78 × 10 3 au. The β vec is the projection of hyperpolarizability on dipole moment vector and shows close resemblance β o. However, good agreement between β o and β vec shows that the direction of the dipole moment vector and the projection of hyperpolarizability are in the same direction. The factor affecting β vec values might be the same for β o , where the highest β vec values are obtained for higher-sized alkali metals (Table 4).

Scattering Hyperpolarizability (β HRS )
Density functional theory calculations have been carried out to find scattering hyperpolarizability (β HRS ), and values range from 1.34 × 10 3 to 1.62 × 10 4 au, where values are increasing steadily from the C1 to C3 cluster. The computed highest value is (1.62 × 10 4 au), found for C1 cluster, whereas the lowest value of 1.34 × 10 3 au is for the C1 cluster ( Table 4). The β HRS is the most viable parameter to calculate the hyperpolarizability of centrosymmetric molecules and clusters, even with zero change in dipole moment. There is an excellent agreement of β HRS with β o where the β HRS show dependence on the size of alkali metals (M). The increased size of alkali metals (Li to K) favors the excellent electronic properties. Therefore, it also causes significantly enhanced β HRS values. Additionally, average dipolar and octupolar hyperpolarizability, which are more prominent in C2 and C3 clusters, provide a notable contribution to β HRS . Moreover, these clusters are of octupolar molecules, which can be seen by their highest octupolar contribution Φβ(j = 3) of 92 % for C3 (Table 4).

Frequency Dependent NLO Properties
We theoretically examined the incident-frequency (ω) effect on the first and second hyperpolarizability at applied frequencies of 532 and 1064 nm. The frequency-dependent first hyperpolarizability β(ω) is calculated with the electro-optical Pockel's effect (EOPE) and second harmonic generation (SHG), whereas the γ(ω) is expressed in terms of dc-Kerr effect and second harmonic generation (SHG). Overall, the dynamic hyperpolarizabilities values are higher than those of static hyperpolarizabilities. The significant EOPE effect β(−ω; ω,0) was observed for the C3 cluster at 532 nm, while its SHG value increased up to 1.7 × 10 6 au (Table 5). It can be demonstrated that the dynamic hyperpolarizabilities are higher at the smaller incident frequency (ω = 532 nm) and slightly decreased at the higher dispersion frequency (1064 nm). Strikingly, the β(ω) values are much more pronounced for the EOPE effect at both frequencies. Table 5. Hyperpolarizability (β 0 in au), frequency-dependent hyperpolarizability β(ω) in terms of electro-optic-Pockel's effect (EOPE) β (−ω; ω, 0) in au, and electric field induced second harmonic generation (EFSHG) β (−2ω; ω, ω) in au at ω = 532 au.

Computational Details
All density functional theory (DFT) calculations are performed in the gas phase with Gaussian 09 software, whereas visualization is achieved using the GaussView 5.0 program [56,57]. Geometries of all polynuclear C 3 O[C(CN) 2 ] 2 M 3 (where M = Li, Na, and K) clusters are optimized at CAM-B3LYP/6-311++G(d,p) functionality [58]. The quantum mechanics-based Coulomb attenuating method (CAM-B3LYP) is a hybrid exchangecorrelation functional that combines B3LYP's hybrid features with the CAM functional's long-range corrected parameter. It was found that this long-range corrected density functional substantially reduces the overestimation seen with conventional techniques and typically provides results that are comparable to those of coupled cluster calculations. Previous research has demonstrated that this method is well recognized for examining molecules and clusters, as well as for determining NLO properties [59,60]. Besides, the choice of a suitable basis set is crucial for obtaining reliable results. Thus, the CAM-B3LYP method with 6-311+G(d,p) split valence basis set is a reliable level of theory for geometry optimization and accuracy in results for electronic properties [61][62][63][64][65].
To determine whether the presented structures are true minima on the potential energy surface, frequency calculations are carried out. For thermodynamic stability, we calculated binding energy per atom for these clusters. Electronic stability and superalkali nature are validated through computed ionization energies (IE) and electron affinities (EA). To further explore the electronic properties, we performed frontier molecular orbital (FMO) analysis. Natural bonding orbitals (NBO) study is carried out to explore the charge distribution on atoms within superalkali clusters [66]. The binding energy per atom (E B ) is given by the following relations: where E T is the total electronic energy of studied (X) superalkali clusters, E A (X) is the total energy of individual atoms within clusters, and n is the total number of atoms. The vertical ionization energy, electron affinity, and electrical conductivity (σ) can be represented by the equation: where VIE and VEA are vertical ionization energies and electron affinities of studied clusters. In Equation (4), σ, E G , k, and T represent the electrical conductivity, energy gap, Boltzmann constant, and temperature, respectively. To further explore the electronic properties of studied clusters, we also performed total density of state (TDOS) analysis at the same method by using the GaussSum software [67]. The following equation can be used to explain the molecules under the static electric field.
where F is an external applied electric field, F i is the component of field along i direction, E 0 is the total energy of the superalkali clusters without a static electric field, and µ i, α ij , β ijk , and γ ijkl are dipole moment, polarizability, hyperpolarizability, and second-order hyperpolarizability, respectively. The mean dipole moment (µ o ), change in dipole moment (∆µ), static polarizability (α o ), and static first hyperpolarizability (β o ) are calculated to illustrate the NLO response and associated responsible factors.
µ o = (µ x 2 + µ y 2 + µ z 2 ) 1/2 (8) To obtain absorption behaviors and excited state parameters of studied clusters, we performed TD-DFT simulations. We considered 30 states for getting excited states parameters. The Gaussian band shape and the absorption spectra were obtained by using the following relation, ϑ: where the i subscript represents the electronic excitation of interest. The other symbols in the equation have the following meanings: • ϑ i , shows the excitation energy (in wavenumbers) corresponding to the required electronic excitation in TD-DFT • ε i max is the value of at the maximum of the band shape • Sigma (σ) is a wavenumber representation of the standard deviation that is related to the simulated band's width.

Conclusions
In summary, we presented the geometric, electronic, and nonlinear optical properties of polynuclear carbon-based clusters at CAM-B3LYP/6-311++G(d,p) level. These clusters are thermodynamically stable, and their binding energies per atom range from −160.07 to −162.07 kcal mol −1 . The electronic stability and superalkali nature are characterized through calculated ionization potential (IP) and FMO analyses. Small ionization potential further suggests their superalkali nature. NBO charge analysis reveals excellent charge separation within clusters. The performed DOS analysis shows size-dependent electronic and conductive properties, where C3 is a potential candidate. The significant first and second hyperpolarizabilities, up to 5.78 × 10 3 and 5.55 × 10 6 au, respectively, are calculated for the clusters. The β o response shows dependence on the size of alkali metals. Furthermore, the absorption study shows their transparency in the deep-UV region for NLO applications and absorption at longer wavelengths. The significant scattering hyperpolarizability (β HRS ) value is (1.62 × 10 4 ), calculated for the C3 cluster, where octupolar contribution to β HRS is 92%. The dynamic first hyperpolarizability β(ω) is more pronounced for the EOPE effect at 532 nm, whereas SHG is more prominent for second hyperpolarizability γ(ω).

Data Availability Statement:
The author confirms that data supporting finding current study are available within article and in its supporting information. Raw data that supports the finding of this study are available from the corresponding author's upon request.